The marginalization paradox and the formal Bayes' law

TL;DR

This paper investigates the marginalization paradox and demonstrates that interpreting improper inferences as probability limits, rather than pointwise limits, resolves the paradox by clarifying the role of the formal Bayes' law.

Contribution

It clarifies the distinction between probability limits and pointwise limits, explaining how this difference resolves the marginalization paradox.

Findings

Probability limits do not necessarily satisfy the formal Bayes' law.

The resolution of the MP depends on interpreting improper inferences as probability limits.

Differences between limits underpin key interpretative issues in Bayesian inference.

Abstract

It has recently been shown that the marginalization paradox (MP) can be resolved by interpreting improper inferences as probability limits. The key to the resolution is that probability limits need not satisfy the formal Bayes' law, which is used in the MP to deduce an inconsistency. In this paper, I explore the differences between probability limits and the more familiar pointwise limits, which do imply the formal Bayes' law, and show how these differences underlie some key differences in the interpretation of the MP.